arXiv:math/0401153v1 [math.SP] 14 Jan 2004 reutberpeetto fS() h smtygopo isometry the SO(4), of representation irreductible oee,teefntosso opcla rprisunder properties peculiar no show functions these However, airudrsm fterttoso O4;ti generalize this SO(4); of the rotations of the of some under havior hc eeaiei oesne(e eo)teuulspheric usual the below) (see sense Y some in generalize which o h iemdso h peia pcso h form the of spaces spherical the of eigenmodes the for SO(4). of rotation ieso ( dimension pnpolm ic hs r iemdsof eigenmodes are those Since problem. open ne h oain fΓ ti la htti erhrequire search u of functions this basis that the clear of is properties rotation it the Γ, of of standing rotations the under where hsnrtto fS() hntertto sa oooyt holonomy an is space rotation spherical the a When of SO(4). mation of rotation chosen a h rnfrainlw ewe hsbssadteuulhyp usual the and basis this harmonics. between laws transformation ve the null involving polynomia harmonic homogeneous peculiar iemdsof eigenmodes h ievle fteLpain∆of ∆ Laplacian the of eigenvalues The Introduction 1 for derivation the le present we of paper, space. eigenmodes forthcoming the a In (re-)deriving space. by method our illustrate ) a ieso ( dimension has 2), eitoueanwbssB,cntutdfo h reductions the from constructed B3, basis new a introduce we he three fayegnoe ne nabtayrtto fS() This of SO(4). functions of those rotation of select arbitrary to properties an possibility under transformation eigenmode, the any explicitely of write to able alca iemdsfrtetresphere three the for eigenmodes Laplacian ℓm hr r w omnyue ae hratrB n 2 for B2) and B1 (hereafter bases used commonly two are There xetdfrsm ae ln n rs pcs e eo) t below), see spaces, prism and (lens cases some for Excepted h ako hsppri oeaietertto rpriso properties rotation the examine to is paper this of task The hnst h utrincrpeettosof representations quaternionic the to Thanks h etrspace vector The o h w-pee h ucin fteebsshv frien a have bases these of functions The two-sphere. the for Y k ℓm ∈ ob iefntoso h nua oetmoeao nIR in operator momentum angular the of eigenfunctions be to IN k evc ’srpyiu,C .Saclay E. C. d’Astrophysique, Service +1) + S 19 i u vtecdx France cedex, Yvette sur Gif 91191 o ie au of value given a For . 3 S orsodn otesm eigenvalue same the to corresponding , 3 2 V / hsvco pc osiue h ( the constitutes space vector This . ,wihrmisa pnpolm ngnrl We general. in probleme open an remains which Γ, k k sapeaaoywr o h erhfreigenfunc- for search the for work preparatory a as , 1) + V k coe 0 2018 10, October fteegnucin fteLpaino the on Laplacian the of eigenfunctions the of .Lachi`eze-ReyM. 2 fe ealn h tnadbssfor bases standard the recalling After . S 3 / Abstract ,ti ie ehdt acltsthe calculates to method a gives this Γ, 1 S k V 3 hysa h eigenspace the span they , k r fteform the of are hc eanivratunder invariant remain which S 3 hc eaninvariant remain which S 3 k n O4,w are we SO(4), and +1) S λ 3 λ k f / drSO(4). nder h property the s tr.W give We ctors. k 2 3 n thus and B3, S ean an remains Γ = sadprism and ns dodecahedral lharmonics al 3 dimensional nunder- an s = er-spherical the . − ffr the offers esearch he k − to ransfor- ( general l be- dly k k V S the f 2), + ( k 3 k V V of of k + 3 k . , tions of S3/Γ (in particular for dodecahedral space). This will be done through the introduction of a new basis B3 of k (in the case k even), V for which the rotation properties can be explicitely calculated: following a new procedure (that was already applied to S2 in [5]) we generate a system of coherent states on k. We extract from it a basis B3 of k, V V which seems to have been ignored in the literature, and presents original k properties. Each function ΦIJ of this basis B3 is defined as [the reduction to S3 of] an homogeneous in IR4, which takes the very simple form (X N)k. Here, the dot product extends the Euclidean · [scalar] dot product of IR4 to its complexification C4, and N is a null vector of C4, that we specify below. After defining these functions, we show that they form a basis of k, and we give the explicit transformation formulae V between B2 and B3. The properties of the basis B3 differ from those of the two other bases, and make it more convenient for particular applications. In particular, it is possible to calculate explicitely its rotation properties, under an arbi- trary rotation of SO(4), by using their quaternionic representation (section 3). This allows to find those functions which remain invariant under an arbitrary rotation. In section 4, we apply these result to (re-)derive the eigenmodes of lens and prism space.

2 Harmonic functions

A function f on S3 is an eigenmode [of the Laplacian] if it verifies ∆f = + λf. It is known that eigenvalues are of the form λk = k (k +2), k IN . − ∈ k The corresponding eigenfunctions generate the eigen[vector]space , of V dimension (k + 1)2, which realizes an irreducible unitary representation of the group SO(4). First basis I call B1 the most widely used basis for k provided by the hyper- V spherical harmonics

B1 ( kℓm Yℓm), ℓ = 1..k, m = ℓ..ℓ. (1) ≡ Y ∝ −

It generalizes the usual spherical harmonics Yℓm on the sphere. In fact, it can be shown ([1], [2] p.240,[3]) that a basis of this type exists on any sphere Sn. Moreover, [2] [3] show that the B1 basis for Sn is “ naturally generated ” by the B1 basis for Sn−1. In this sense, the B1 basis for S3 2 is generated by the usual spherical harmonics Yℓm on the 2-sphere S . The generation process involves harmonic polynomials constructed from null complex vectors (see below). The basis B1 is in fact based on the reduction of the representation of SO(4) to representations of SO(3): each kℓm is an of an SO(3) of SO(4) which leaves Y a selected point of S3 invariant. This make these functions useful when one considers the action of that peculiar SO(3) subgroup. But they show no simple behaviour under a general rotation. We will no more use this basis. Second basis By group theoretical arguments, [1] construct a different ON basis of V k, which is specific to S3:

B2 (Tk;m ,m ), m1,m2 = k/2...k/2, (2) ≡ 1 2 − where m1 and m2 vary independently by entire increments (and, thus, take entire or semi-entire values according to the of k). In the

2 spirit of the construction refered above, B2 may be seen as generated from a different choice of spherical harmonics on S2. The bases B1 and B2 appear respectively adapted to the systems of hyperspherical and toroidal (see below) coordinates to describe S3. The formula (27) of [1], reduced to the three-sphere, shows that the elements of this basis take a very convenient form if we use toroidal coordi- nates (as they are called by [7]) on the three sphere S3: (χ,θ,φ) spanning the range 0 χ π/2, 0 θ 2π and 0 φ 2π. They are conve- ≤ ≤ ≤ ≤ ≤ ≤ niently defined (see [7] for a more complete description) from an isometric embedding of S3 in IR4 (as the hypersurface x IR4; x = 1): ∈ | | x0 = r cos χ cos θ x1 = r sin χ cos φ  x2 = r sin χ sin φ   x3 = r cos χ sin θ µ  4 where (x ), µ = 0, 1,2, 3, is a point of IR . As shown in [7]), these coordinates appear naturally associated to some isometries. Very simple manipulations show that, with these coordinates, each eigenfunction of B2 takes the form: iℓθ imφ Tk;m ,m (X) tk;m ,m (χ) e e , (3) 1 2 ≡ 1 2 where the tk;m1,m2 (χ) are polynomials in cos χ and sin χ and we wrote, for simplification, ℓ m1 + m2, m m2 m1. ≡ ≡ − To have a convenient expression, we report this formula in the har- monic equation expressed in coordinates χ,θ,φ. This leads to a second order differential equation (cf. equ 15 of [7]). The solution is proportional ℓ m m,ℓ to a Jacobi polynomial: tk;m ,m (χ) cos χ sin χ P (cos 2χ), d 1 2 ∝ d ≡ k/2 m2. Thus, we have the final expression for the basis B2 − iθ ℓ iφ m (m,ℓ) Tk;m1,m2 (X)= Ck;m1,m2 [cos χ e ] [sin χ e ] Pd [cos(2χ)], (4)

√(k+1) (k/2+m2)! (k/2−m2)! with Ck;m ,m − from normalization re- 1 2 ≡ π (k/2+m1)! (k/2 m1)! quirements (the variation rangesq of m1 anf m2 imply that the quantities under factorial sign are entire and positive). Note also the useful propor- tionality relations: ℓ m (m,ℓ) ℓ −m (−m,ℓ) cos χ sin χ P k−ℓ−m) (cos 2χ) cos χ sin χ P k−ℓ+m) (cos 2χ) 2 ∝ 2 −ℓ m (m,−ℓ) cos χ sin χ P k+ℓ−m (cos 2χ). ∝ 2 The term ζm ξn eiℓθ eimφ in (4) defines the rotation properties of ≡ Tk;m1,m2 under a specific subgroup of SO(4). This properties generalizes the properties of the spherical harmonics on the two-sphere S2, to be eigenfunctions of the rotation operator Px. This advantage has been used by [7] to calculate (from a slightly different basis) the eigenmodes of lens or prism spaces (see below section 4). However, the Tk;m1,m2 have no simple rotation properties under the general rotation of SO(4). This motivates the search for a different basis of k. V Note that the basis functions Tk;m1,m2 have also been introduced in [2] (p. 253), with their expression in Jacobi Polynomials. Note also that they are the complex counterparts of those proposed by [7] (their equ. 19) to find the eigenmodes of lens and prism spaces. The variation range of the indices m1,m2 here (equ. 2) is equivalent to their condition ℓ + m k, ℓ + m = k, mod (2), (5) | | | |≤ through the correspondence ℓ = m1 + m2, m = m2 m1. −

3 2.1 Complex null vectors A complex vector Z (Z0, Z1, Z2, Z3) is an element of C4. We extend ≡ the Euclidean scalar product in IR4 to the complex (non Hermitian) inner ′ µ ′ µ null product Z Z µ Z (Z ) , µ = 0, 1, 2, 3. A vector N is defined · ≡ µ µ as having zero norm N N N N = 0 (in which case, it may be P · ≡ µ considered as a point on the isotropic cone in C4). It is well known that P polynomials of the form (X N)k, homogeneous of degree k, are harmonic · if and only if N is a null vector. This results from

k k k−1 ∆0(X N) ∂µ ∂µ (X N) = k (Nµ Nµ) (X N) = 0, · ≡ · · µ µ ! X X 4 where ∆0 is the Laplacian of IR . Thus, the restrictions of such polyno- mials belong to k. As we mentioned above, peculiar null vectors have V been used in [2] and [3] to generate the bases B1 and B2. To construct a third basis B3, let us first define a family of null vectors

N(a,b) (cosa, i sin b, i cos b, sin a), (6) ≡ indexed by two angles a and b describing the unit circle (they define coherent states in IR4). The polynomial [X N(a,b)]k is harmonic and, thus, can be decom- · posed onto the basis B2. It is easy to check that, like the scalar product X N(a,b), this polynomial depends on a and b only through the com- · i(θ−a) i(φ+b) binations e and e , with their conjugates. This implies that its decomposition on B2 takes the form

k −ia(m1+m2) ib(m2−m1) [X N(a,b)] = Pk;m ,m Tk;m ,m (X) e e , · 1 2 1 2 m ,m X1 2 (7) where the coefficients Pk;m1,m2 do not depend on a,b. Now we intend to find a basis of k in the form of such polynomials. V 2.2 An new basis 2.2.1 Roots of unity To do so, we consider the (k + 1)th complex roots of unity which are the powers ρI of

2iπ 2π ρ e k+1 cos α + i sin α, α . (8) ≡ ≡ ≡ k + 1 We recall the fundamental property, which will be widely used thereafter:

k nI Dirac ρ =(k + 1) δI , (9) n=0 X where the equallity in the Dirac must be taken mod k + 1. In a given frame, we consider the family of null vectors

NIJ N(Iα, Jα) = (cos Iα, i sin Jα, i cos Jα, sin Iα), I,J = 0..k ≡ (10) k k k and we define the functions Φ : Φ (X) (X NIJ ) . We report such IJ IJ ≡ · a function in equ.(7) to obtain its development in the basis B2. Then we − − multiply both terms by ρI(m1+m2) J(m2 m1). Making the summations

4 over I, J (each varying from 0 to k), and using (9), we obtain, in the case where k is even (that we assume hereafter):

k 1 I(m1+m2)−J(m2−m1) k k;m ,m = ρ Φ , (11) T 1 2 (k + 1)2 IJ I,J=0 X where k;m ,m Pk;m ,m Tk;m ,m . T 1 2 ≡ 1 2 1 2 This gives the decomposition of any Tk;m1,m2 (and thus, of any har- 2 k monic function) as a sum of the (k + 1) polynomials ΦIJ , providing the new basis of k: V B3 (Φk ), I,J = 0..k (k even). (12) ≡ IJ

The coefficients Pk;m1,m2 involved in the transformation are calculated in Appendix A. We obtain easily the reciprocal formula expressing the change of basis:

k/2 k −I(m1+m2)+J(m2−m1) ΦIJ = k;m1,m2 ρ . (13) − T m1,mX2= k/2 3 Rotations in IR4 3.1 Matrix representations The isometries of S3 are the rotations in the embedding space IR4. In the usual matrix representation, a rotation is represented by a 4 4 orthogonal ∗ matrix g SO(4), acting on the 4-vector (xµ) by matrix product. ∈ 3 In the complex matrix representation, a point (vector) of IR is repre- sented by the 2 2 complex matrix ∗ W iZ X ; W x0 + ix3, Z x1 + ix2 C. ≡ iZ¯ W¯ ≡ ≡ ∈  

A rotation g is represented by two complex 2 2 matrices (GL, GR), so ∗ that its action takes the form X GL X GR (matrix product). The two 7→ 3 matrices GL and GR belong to SU(2). Since SU(2) identifies with S , any matrix GL or GR is of the same form than the matrix X above. Since SU(2) is also the set of unit norm , there is a quaternionic representation for the action of SO(4).

3.2 Quaternionic notations

Let us note jµ, µ = 0, 1, 2, 3 the basis of quaternions (the jµ correspond to the usual 1, i, j, k but we do not use this notation here). We have j0 = 1. µ 0 i A general is q = q jµ = q +q ji (with summation convention; the index i takes the values 1,2,3; the index µ takes the values 0,1,2,3). 0 i Its quaternionic conjugate isq ¯ q q ji. The scalar product is q1 q2 ≡ − 2 qq¯ µ ·2 ≡ (q1 q¯2 + q2 q¯1)/2, giving the quaternionic norm q = 2 = µ(q ) . µ 4 | | µ We represent any point x =(x ) of IR by the quaternion qx x jµ. 3 P ≡ 2 The points of the (unit) sphere S correspond to units quaternions, q = | | 1. Hereafter, all quaternions will be unitary (if no otherwise indicated). It is easy to see that, using the coordinates above, a point of S3 is represented by the quaternion cos χ ζ˙ + sin χ ξ˙ j1, where we define dotted quantities, like ζ˙ cos θ + j3 sin θ, ξ˙ cos φ + j3 sin φ, as the quaternionic analogs ≡ ≡

5 of the complex numbers ζ = cos θ + i sin θ and ξ = cos φ + i sin φ, i.e., with the imaginary i replaced by the quaternion j3. In quaternionic notation, the rotation g : x gx is represented by a 7→ pair of unit quaternions (QL,QR) such that qx qgx = QL qx QR. Complex quaternions, null quaternions7→ The null vectors N introduced above do not belong to IR4 but to C4. Thus, they cannot be represented by quaternions, but by complex quaternions. Those are defined exactly like the usual quaternions, but with complex rather than real coefficients. Note that the pure imaginary i does not coincide with any of the jµ, but commutes with all of them. Also, complex conjugation (star) and quaternionic conjugation (bar) must be carefully distinguished. Then it is easy to see that the (null) vectors NIJ I J defined above correspond to the complex quaternion nIJ ρ˙ + i j2 ρ˙ . 2 ≡ Note that nIJ = 0. | | In quaternionic notations, the basis functions are expressed as

k k k nIJ q¯X + qX n¯IJ ΦIJ (x)=(NIJ x) =< nIJ qX > = . (14) · · 2   Quaternionic notations will help us to check how the basis functions are transformed by the rotations of SO(4).

3.3 Rotations of functions

To any rotation g, is associated its action Rg on functions: Rg : f 7→ Rgf; Rg f(x) f(gx). Let us apply this action to the basis functions: ≡ k g ΦIJ (x) = ΦIJ (gx)=< nIJ (QL qx QR) > . (15) R · We consider a function on S3 also as a functions on the set of unit quater- 3 nions (qx is the unit quaternion associated to the point x of S ). On the other hand, we may develop this function on the basis:

k ij RgΦIJ G (g) Φij . (16) ≡ IJ ij=0 X ij The coefficients GIJ (g) of the development, that we intend to calculate, completely encode the action of the rotation g on the basis B3, and thus on V k. To proceed , we introduce three auxiliary complex quaternions:

α 1+ i j3, β j1 i j2 = (1 i j3) j1 and δ j1 i j2. ≡ ≡ − − ≡− − I They have zero norm and obey the properties < α nIJ >= ρ , −I J · −J < α¯ nIJ >= ρ , < β nIJ >= ρ , < δ nIJ >= ρ . Let us now · · · estimate the relation (16) for the specific quaternion α + R α¯ + S β + T δ, with R,S,T arbitrary real numbers:

′ − − ( +R +S +T )k = Gij (g) < (ρi+Rρ i+Sρj +T ρ j )k, (17) A A B D IJ ij X ′ where , , , characterize the rotation. (Note that these D ≡ · quantities depend on I and J). We develop and identify the powers of the exponents R,S,T :

′ − − − − − q p q r k p r = Gij (g) ρi(2q p) ρj(2r k+p). A A B D IJ ij X 6 This holds for 0 q p, 0 r k p, 0 p k. After definition of ≤ ≤ ≤ ≤ − ≤ ≤ the new indices A q + r, B q r + k p, which both vary from 0 to ≡ ≡ − − k, the previous equation takes the form

A/2 B/2 ′ p/2 ′ k/2 ij i(A+B−k)+j(A−B) AB′ AD′ AA A BD = GIJ (g) ρ . ij A D  A B   BD   A  X This holds for any value of A,B,p. A consequence is that ′ = , AA BD which can be checked directly. Finally, ′ − − A B k = Gij (g) ρi(A+B k) ρj(A B), U V A IJ ij  X B A with ′ , . U ≡ A V ≡ B Taking into account the properties of the roots of unity, this equation   has the solution

′ k k ( ) − − − − Gij = A ρ i(A+B k) ρ j(A B) A B . (18) IJ (k + 1)2 U V A,B=0 X When a rotation is specified, there is no difficulty to estimate the associ- ated values of ′, , , and thus of these coefficients which completely A U V encode the transformation properties of the basis functions of V k under SO(4). In the next section, we apply these results to rederive the eigenmodes of Lens or Prism space. In the next paper [6], we take for g the generators of Γ, the group of holonomies of the dodecahedral space. This will allow the selection of the invariant functions, which constitute its eigenmodes.

4 Lens and Prism space

The eigen modes for Lens and Prism space have been found by [7]. Here we derive them again for illustration of our method.

4.1 Lens space An holonomy transformation of a lens space takes the form, in complex notation,

ψ +ψ ψ −ψ i 1 2 i 1 2 e 2 0 e 2 0 GL = ψ +ψ , GR = ψ −ψ ) G. −i 1 2 −i 1 2 " 0 e 2 # " 0 e 2 # (19) Its action on a vector of IR4 takes the form W iZ Weiψ1 iZeiψ2 X GL X GR = − −iψ . (20) ≡ iZ¯ W¯ 7→ iZe¯iψ2 We¯ 1     In this simple case, W x0 + ix3, Z x1 + ix2 are transformed into iψ iψ ≡ ≡ W e 1 and Z e 2 respectively. This corresponds to the quaternionic notation

QL =w ˙ 1 w˙ 2,QR =w ˙ 1/w˙ 2, w˙ i cos(ψi/2) + j3 sin(ψi/2). (21) ≡ The rotation is expressed in the simplest way in the toroidal coordi- nates, since it acts as θ θ + ψ1, φ φ + ψ2. From the expression (4) 7→ 7→ of the basis functions (B2), it result their transformation law :

ℓψ1+mψ2 Rg : Tk;m ,m Tk;m ,m e . 1 2 7→ 1 2

7 This leads directly to the invariance condition ℓψ1 + mψ2 =0 mod2π. Using the standard notation for a lens space L(p, q), namely

ψ1 = 2π/p, ψ2 = 2π q/p, we are led to the conclusion:

the eigenmodes of lens space L(p, q) are all linear combinations of Tk;m1,m2 , where the underlining means that the indices verify the condition m1 + m2 + q(m2 m1) = 0, modulo(p). − 4.2 Prism space

The two generators are single action rotations (GR = 0). The first gener- ator, analog to the lens case above, with ψ1 = ψ2 = 2π/2P , provides the first condition ℓ + m = 0, mod 2P which takes the form

m2 =0 mod P. (22)

This implies that k must be even. 0 i The second generator has the complex matrix form G = GL = − , i 0  −  which corresponds to the quaternion QL = Q = j1. Easy calculations ′ − − − lead to = ρJ , = ρ J , = ρI , = ρ I . Reporting in (18) gives A − A B D − (i−J) k k ρ − − − − Gij = ρA ( i j+I+J)+B ( i+j I+J) ( 1)B . (23) IJ (k + 1)2 − A,BX=0 This formula, together with those expressing the change of basis between B2 and B3, allow to return to the rotation properties of the basis B2 which take the simple form:

m2+k/2 R : k;m ,m ( 1) k;m ,−m . (24) T 1 2 7→ − T 1 2 It results immediately that the G-invariant functions are combinations of m2+k/2 k;m ,m +( 1) k;m ,−m . T 1 2 − T 1 2 Finally,

the eigenfunctions of the Prism space are combinations of m2+k/2 k;m ,m +( 1) k;m ,−m , m1; k even. T 1 2 − T 1 2 ∀ According to the parity of k/2, the functions k;m ,0 are included or not, T 1 from which simple counting give the multiplicity as (k +1) (1+[k/2P ]), for k even ([...] means entire value), (k + 1) [k/2P ], for k odd, in accordance with [4].

5 Conclusion

We have shown that V k, the space of eigenfunctions of the Laplacian of S3 with a given eigenvalue λk (k even) admits a new basis B3. In contrary to standard bases (B1 and B2) which show specific rotation properties under selected of SO(4), it is possible to calculate explicitely the rotation properties of B3 under any rotation of the group SO(4), as well as to calculate the functions invariant under this rotation. This opens the door to the calculation of eigenmodes of spherical space. The

8 eigenfunctions of lens and prism spaces had been calculated by [7], by using a basis related to B2 (its real, rather than complex, version). We rederived them to illustrate the properties of the bases. In a subsequent paper [6], we apply these results to the search of 3 ∗ the eigenfunctions of the dodecahedral space S /Γ, where Γ = DP is the binary dihedral group of order 4P . Those functions, still presently unknown, are the eigenfunctions of S3 which remain invariant under the elements of Γ.

5.1 Appendix A Let us evaluate the function

k − Zk (X) ρℓI Jm Φk (X) (25) ℓm ≡ IJ IJX=0 k − − − 1 1 = 2 k ρℓI mJ cos χ (ζρ I + ) + sin χ (ξρJ ) , ζρ−I − ξρJ XIJ   where we defined ζ eiθ and ξ eiφ. After development of the power ≡ ≡ with the binomial coefficients, the sum becomes

k − k − 1 − 1 ρℓI mJ [cos χ (ζρ I + )]k p [sin χ (ξρJ )]p. (26) p ζρ−I − ξρJ p=0 XIJ X   Let us write the identities

k−p − 1 − k p − − − − ρℓI (ζρ I + )k p = − ζ2r+p k ρ I(2r+p k ℓ), (27) ζρ−I r r=0 X  

p − 1 p − − − − ρ mJ (ξρJ )p = ξ2q p( 1)p q ρJ(2q p m), (28) − ξρJ q − q=0 X   that we insert into (26). After summing over I, J, and rearranging the terms, we obtain:

k q−m k−2q+m 2q−m −k ℓ m ( 1) (cos χ) (sin χ) Zℓm(X) = 2 ζ ξ k! − k+ℓ−2q+m k−ℓ−2q+m . (29) q q! (q m)! ( 2 )! ( 2 )! X − This formula results from the fact that, through (9), the summations over I, J imply p = 2q m and 2r = ℓ + k + m 2q, that we have reported. − − The range of the summation over q is defined by the conditions

0 ℓ + k + m 2q 2k + 2m 4q 2k, 0 q 2q m k. (30) ≤ − ≤ − ≤ ≤ ≤ − ≤ Rearrangements of the previous formula, inserting u cos(2χ) = 2 2 ≡ 2 cos χ 1 = 1 2 sin χ, lead to − − −3k/2 ℓ m ℓ m 2 ζ ξ k! (1+ u) 2 (1 u) 2 Zℓm(X)= − (31) (m + d)! (ℓ + d)!

m + d ℓ + d − (1 + u)i (u 1)d i, i d i − q X    − 

9 where we have defined i k+m−ℓ q and d k−ℓ−m . Verification shows ≡ 2 − ≡ 2 that the range defined as above gives exactly the development formula for the Jacobi polynomial. The comparison with (11) gives the coefficient

2−k k! Pk;m1,m2 = 2 (k/2 m1)! (k/2+ m1)! (k + 1) Ck;m ,m − 1 2 2−k π k! (k + 1)−5/2 = (k/2+ m2)! (k/2 m2)!(k/2+ m1)! (k/2 m1)! − − p References

[1] and the (I), Bander M. & Itzykson C., Rev. Mod. Phys. 18,2,1966 [2] Higher transcendental Functions A. Erd´elyi, W. Magnus, F. Ober- hettinger, F. G. Tricomi, McGraw-Hill 1953 [3] A. Fryant, SIAM J. Math. Anal., Vol. 22, N 1, pp. 268-271, 1991 [4] Ikeda A. 1995, Kodai Math . J. 18 (1995) 57-67 [5] Lachi`eze-Rey M. 2003, Journal of Physics A: Mathe- matical and General, volume 37, issue 1, pages 205 - 210(http://fr.arxiv.org/abs/math.SP/0304409) [6] Lachi`eze-Rey M. 2004, Eigenmodes of dodecahedral space, in prepa- ration [7] Lehoucq R. Uzan J. P., Weks J. 2002, math.SP/0202072 [8] Fonctions sp´eciales de la physique math´ematique, A. Nikiforov, V. Ouvarov, Editions Mir (Moscou) 1983

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